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A186008 Irregular triangle T(n,k) read by rows, in which row n has the pattern of conjectured dropping times in the Collatz iteration.

%I A186008 #24 Mar 30 2012 17:22:57
%S A186008 2,4,16,12,32,8,52,128,40,56,84,136,160,180,256,60,80,136,220,288,296,
%T A186008 448,528,636,688,712,1024,152,232,384,648,704,788,856,1000,1204,1416,
%U A186008 1472,1556,1592,1624,1800,1972,2008,2120,2356,2360,2676,2744,2888,2912,3064,3328,3444,3680,3832,4096
%N A186008 Irregular triangle T(n,k) read by rows, in which row n has the pattern of conjectured dropping times in the Collatz iteration.
%C A186008 Consider A126241, the sequence of dropping times in the Collatz iteration. Only zero and the numbers in A020914 can be dropping times. The dropping times in A126241 have a definite pattern. For example, 1 appears at positions n = 2 + 2*i, for i=0,1,2,3,... Similarly, 2 appears at positions n = 5 + 4*i; 4 appears at n = 3 + 16*i; 5 appears at n = 11 + {12,32}*i; and 7 appears at 7 + {8, 52, 128}*i. In general, if we let s=A020914(r) be the r-th possible stopping time, then A126241(n) = s for n = A122442(r) + T(r)*i, where T(r) is the r-th row of this triangle.  The length of row n is A186009(n). The n-th row ends with 2^A020914(n).
%C A186008 The frequency of the r-th dropping time s=A020914(r) can be computed as A186009(r)/2^s. The first few frequencies are 1/2, 1/4, 1/16, 1/16, 3/128, 7/256, 3/256, 15/2048, and 85/8192.
%C A186008 The term "stopping time" is sometimes used instead of "dropping time", but the former usually refers to A006666.
%C A186008 This sequence is closely related to A177789.
%D A186008 J. C. Lagarias, ed., The Ultimate Challenge: The 3x+1 Problem, Amer. Math. Soc., 2010. See pp. 33, 35ff.
%H A186008 <a href="/index/3#3x1">Index entries for sequences related to 3x+1 (or Collatz) problem</a>
%e A186008 The triangle begins
%e A186008 2
%e A186008 4
%e A186008 16
%e A186008 12, 32
%e A186008 8, 52, 128
%e A186008 40, 56, 84, 136, 160, 180, 256
%e A186008 60, 80, 136, 220, 288, 296, 448, 528, 636, 688, 712, 1024
%Y A186008 Cf. A014682, A126241, A006666, A186009.
%K A186008 nonn,tabf
%O A186008 1,1
%A A186008 _T. D. Noe_, Feb 09 2011